When a goat damages one of its horns, it can be a very painful experience. There is a blood vessel and a nerve that extends up into each horn, so depending on the extent of the injury, there can be a lot of bleeding. A wound that is not life threatening can still produce what looks like a large amount of blood, so the injury might not be as bad as you think. Prompt first aid should given, however, just as with any other injury. Baby aspirin can be given to the goat for pain relief.
If the horn has a broken tip that isn't bleeding or is bleeding very little, then the injury isn't deep enough to require serious first aid. Snip off the broken tip if it is still attached and snip off any jagged edges. Apply a blood-stop powder to stifle the minor bleeding (such powders are available at most pet and farm-supply stores). You may wrap the end of the horn if it is warranted to keep the tip clean while it heals.
Bleeding will occur if the horn is broken down far enough to open the large blood vessel inside. Treat it as any other open wound. Apply pressure to the stump of the horn to control bleeding. Once the bleeding is under control, apply a blood stop powder to encourage clotting and then wrap the horn with a bandage.
Find the pressure point below the inside corner of the goat's eye if direct pressure does not stop the bleeding from the broken horn. Move your finger around a bit in the area to feel for this blood vessel. Push on the pressure point for five minutes while holding direct pressure on the horn stump. Gently release the pressure on the pressure point after five minutes and check the bleeding. Reapply pressure again if blood flow has not ceased. Once the bleeding is under control, apply a blood-stop powder to encourage clotting and then wrap the horn with a bandage.
A red-hot piece of iron or a disbudding iron (a tool used to de-horn young animals) can be used to cauterize the wound if the bleeding is severe and does not respond to other control methods. Get a helper to assist by holding the goat, or put the goat into a stanchion to hold it still while you apply the iron to cauterize the wound. Apply the iron firmly and decisively. Hold it on the wound for a second or two at a time. You want to close the wound, but not burn healthy tissue. Do not apply cream or other first aid remedies to the cauterized area.